Hannah reads at a constant rate of $3$ pages every $8$ minutes. Write an equation that shows the relationship between $p$, the number of pages she reads, and $m$, the number of minutes she spends reading.
Solution: Let's find the constant of proportionality. In the proportional relationship between $p$, the number of pages Hannah reads, and $m$, the number of minutes she spends reading, one constant of proportionality is her reading speed in number of pages per minute. It is the number we multiply by the number of minutes to get the number of pages. $m\,\times\, ?=p$ $\begin{aligned} m\,\times\, {?}&=p \\\\ {?}&=\dfrac{p}{m} \\\\ &={\dfrac{3}{8}} \end{aligned}$ The constant of proportionality is ${\dfrac{3}{8}}$. This means we can multiply ${\dfrac{3}{8}}$ by the number of minutes to get the number of pages. Now, let's write the equation: $\begin{aligned} \text{number of pages}&={\text{reading speed}}\times\text{number of minutes} \\\\ p&={\dfrac{3}{8}}m \end{aligned}$ One correct equation is: $p = \dfrac 38 m$